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David Roberts ♦
David Roberts ♦
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Many years ago, Lawvere showed that the forgetful functor $U: \mathbf{Endo}\to \mathbf{Set}$ has a left adjoint $F$ if and only if $\mathbf{Set}$ has a natural numbers object, where $\mathbf{Endo}$ is the category of endos $f: A\to A$, where $A$ is a set. Then $T=UF$ is the functor part of a monad $(T,\mu,\eta)$. I learned some years ago too that $\mathbf{Endo}$ satisfies the Beck conditions in Beck's Theorem, so that the category of $\mathbf{Set}^T$ of $T$-algebras is equivalent to $\mathbf{Endo}$.

An exercise in Barr and Wells states that under certain conditions, the equivalence guaranteed by Beck's Theorem is in fact an isomorphism (see Exercise PPTT p. 116). My question is: whether, in the case of the monad $(T,\mu,\eta)$ mentioned above, $\mathbf{Endo}$ turns out to be isomorphic to $\mathbf{Set}^T$? --- either because the conditions mentioned in the exercise are satisfied or for some other reason.

Many years ago, Lawvere showed that the forgetful functor $U: \mathbf{Endo}\to \mathbf{Set}$ has a left adjoint $F$ if and only if $\mathbf{Set}$ has a natural numbers object, where $\mathbf{Endo}$ is the category of endos $f: A\to A$, where $A$ is a set. Then $T=UF$ is the functor part of a monad $(T,\mu,\eta)$. I learned some years ago too that $\mathbf{Endo}$ satisfies the Beck conditions in Beck's Theorem, so that the category of $\mathbf{Set}^T$ of $T$-algebras is equivalent to $\mathbf{Endo}$.

An exercise in Barr and Wells states that under certain conditions, the equivalence guaranteed by Beck's Theorem is in fact an isomorphism (see Exercise PPTT p. 116). My question is whether, in the case of the monad $(T,\mu,\eta)$ mentioned above, $\mathbf{Endo}$ turns out to be isomorphic to $\mathbf{Set}^T$ --- either because the conditions mentioned in the exercise are satisfied or for some other reason.

Many years ago, Lawvere showed that the forgetful functor $U: \mathbf{Endo}\to \mathbf{Set}$ has a left adjoint $F$ if and only if $\mathbf{Set}$ has a natural numbers object, where $\mathbf{Endo}$ is the category of endos $f: A\to A$, where $A$ is a set. Then $T=UF$ is the functor part of a monad $(T,\mu,\eta)$. I learned some years ago too that $\mathbf{Endo}$ satisfies the Beck conditions in Beck's Theorem, so that the category of $\mathbf{Set}^T$ of $T$-algebras is equivalent to $\mathbf{Endo}$.

An exercise in Barr and Wells states that under certain conditions, the equivalence guaranteed by Beck's Theorem is in fact an isomorphism (see Exercise PPTT p. 116). My question is: whether, in the case of the monad $(T,\mu,\eta)$ mentioned above, $\mathbf{Endo}$ turns out to be isomorphic to $\mathbf{Set}^T$? --- either because the conditions mentioned in the exercise are satisfied or for some other reason.

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P. Corazza
P. Corazza
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Beck's Theorem and the category of endos

Many years ago, Lawvere showed that the forgetful functor $U: \mathbf{Endo}\to \mathbf{Set}$ has a left adjoint $F$ if and only if $\mathbf{Set}$ has a natural numbers object, where $\mathbf{Endo}$ is the category of endos $f: A\to A$, where $A$ is a set. Then $T=UF$ is the functor part of a monad $(T,\mu,\eta)$. I learned some years ago too that $\mathbf{Endo}$ satisfies the Beck conditions in Beck's Theorem, so that the category of $\mathbf{Set}^T$ of $T$-algebras is equivalent to $\mathbf{Endo}$.

An exercise in Barr and Wells states that under certain conditions, the equivalence guaranteed by Beck's Theorem is in fact an isomorphism (see Exercise PPTT p. 116). My question is whether, in the case of the monad $(T,\mu,\eta)$ mentioned above, $\mathbf{Endo}$ turns out to be isomorphic to $\mathbf{Set}^T$ --- either because the conditions mentioned in the exercise are satisfied or for some other reason.